Chapter 1.1, Problem 15ES

a.

To determine

The fourth polynomial $P_4\left(x\right)$ for the function $f\left(x\right)=x{e}^{x^2}$ about $x_0=0$ and then to use find an upper bound for $\left|f\left(0.4\right)-P_4\left(0.4\right)\right|$ for $0\le x\le 0.4$.

b.

To determine

The value of ${\displaystyle \underset{0}{\overset{0.4}{\int }}f\left(x\right)}dx$, using ${\displaystyle \underset{0}{\overset{0.4}{\int }}{P}_4\left(x\right)}dx$.

c.

To determine

An upper bound for the error in (b) using ${\displaystyle \underset{0}{\overset{0.4}{\int }}{P}_4\left(x\right)}dx$.

d.

To determine

To approximate: The value of $f^{\prime }\left(0.2\right)$ using $P_4{}^{\prime }\left(0.2\right)$ and to find the error.